---
tag: 1VC7EJ1ZBZ
kind: concept
name: Commutant
msc: [46L10, 47L05]
sources:
  - url: https://en.wikipedia.org/wiki/Centralizer_and_normalizer
---

## Summary

Let $S$ be a set of operators on a Hilbert space.
Its commutant $S'$ is the set of all bounded operators that commute with every element of $S$.
It is the operator version of the centralizer in an algebra.
The commutant of a self-adjoint set is always a [[von Neumann algebra|PSFWQVZ0FS]].
By the [[bicommutant theorem|BSA6GFT8SQ]], the von Neumann algebras are the self-adjoint unital algebras equal to their double commutant.

## Explanations

### L1 human

Given some operators, ask which operators can be applied before or after them with the same result.
These form the commutant.
For example, the only $n \times n$ matrices that commute with all $n \times n$ matrices are the multiples of the identity.
In the other direction, the matrices that commute with all diagonal matrices are exactly the diagonal matrices.
So a big set has a small commutant, and a small set has a big commutant.